Take Portfolio A in the plot and suppose it's just a single stock, driven by the GBM above with instantaneous rate of return $\mu$. Portfolio A has an "expected return" of $8\%$. So, which of the following (if any) do we mean?

1 Answer
1

Suppose we have no dividends like in Black-Scholes-Merton and in your example. Expected return between time $t$ and $t+\Delta t$ is defined as
$$
\mathbb{E}_t\left[R_{t+\Delta t}\right]\equiv\mathbb{E}_t\left[\frac{S_{t+\Delta t} - S_t}{S_t}\right] = \mathbb{E}_t\left[\frac{\Delta S_t}{S_t}\right]
$$
You can see that, as $\Delta t \to dt$, $\mathbb{E}_t\left[\frac{\Delta S_t}{S_t}\right] \to \mathbb{E}_t\left[\frac{dS_t}{S_t}\right] = \mu dt$.

Regarding your question, $\mu$ is the instantaneous rate of return so $e^\mu - 1 = 8\%$. Alternatively, you know that rate of returns should be defined for a certain timeframe (e.g. 4% per semester, 8% per year etc…) and with some compounding method (e.g. yearly compounding, quarterly compounding etc…).
$\mu$ is the annual nominal expected interest rate obtained by continuous compounding.